Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 11 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the units digit of 8^11? (a) 8 (b) 6 (c) 2 (d) 4
2. 2.What is the smallest number you must add to 206 to make it divisible by 3? (a) 2 (b) 1
3. 3.Which of these numbers is divisible by 9? (a) 7290 (b) 7287 (c) 7300 (d) 7292
4. 4.What is the remainder when 270 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
5. 5.Which of these numbers is divisible by 3? (a) 1040 (b) 1036 (c) 1033 (d) 1035
6. 6.What is the remainder when 259 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
7. 7.What is the units digit of 4^18? (a) 7 (b) 5 (c) 6 (d) 4
8. 8.What is the units digit of 3^22? (a) 9 (b) 7 (c) 1 (d) 3
9. 9.What is the smallest number you must add to 122 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 1766 (b) 1763 (c) 1764 (d) 1769
11. 11.What is the units digit of 7^42? (a) 1 (b) 7 (c) 3 (d) 9
12. 12.What is the remainder when 364 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
13. 13.What is the remainder when 871 is divided by 11? (a) 9 (b) 3 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 536 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
15. 15.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
16. 16.Which of these numbers is divisible by 4? (a) 2889 (b) 2892 (c) 2899 (d) 2893
17. 17.What is the units digit of 9^49? (a) 8 (b) 9 (c) 1 (d) 7
18. 18.What is the remainder when 744 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 3
19. 19.What is the units digit of 8^25? (a) 6 (b) 2 (c) 8 (d) 4
20. 20.What is the smallest number you must add to 98 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the remainder when 91 is divided by 3? (a) 1 (b) 0 (c) 2
22. 22.Which of these numbers is divisible by 3? (a) 2436 (b) 2435 (c) 2437 (d) 2441
23. 23.What is the units digit of 9^50? (a) 0 (b) 2 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 446 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 1
25. 25.What is the smallest number you must add to 194 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the smallest number you must add to 238 to make it divisible by 4? (a) 3 (b) 2 (c) 1
27. 27.What is the units digit of 4^24? (a) 6 (b) 5 (c) 7 (d) 4
28. 28.What is the units digit of 2^32? (a) 4 (b) 6 (c) 2 (d) 8
29. 29.Which of these numbers is divisible by 3? (a) 2233 (b) 2231 (c) 2228 (d) 2229
30. 30.What is the smallest number you must add to 119 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 4^31? (a) 3 (b) 5 (c) 6 (d) 4
32. 32.Which of these numbers is divisible by 8? (a) 4993 (b) 5014 (c) 5007 (d) 5000
33. 33.Which of these numbers is divisible by 3? (a) 484 (b) 488 (c) 482 (d) 483
34. 34.What is the smallest number you must add to 272 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
35. 35.Which of these numbers is divisible by 3? (a) 845 (b) 842 (c) 847 (d) 843
36. 36.What is the remainder when 76 is divided by 3? (a) 2 (b) 1 (c) 0
37. 37.What is the smallest number you must add to 946 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
38. 38.What is the remainder when 242 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
39. 39.Which of these numbers is divisible by 3? (a) 2258 (b) 2260 (c) 2259 (d) 2264
40. 40.What is the remainder when 73 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2

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